SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC. 
245 
as the present value of the marginal productivity of capital at 
time ¢ multiplied by the marginal utility of consumption at 
that time. Finally, we denote by 
(23) 
Oto) = Üx(s) , where Ûnte, _ 
et {n(x 
3 ~ Wdtfor T < oo 
the utility of the path (£, 2,) under scrutiny. 
Propositions (F) and (G) hold for any eligible-attainable 
path (£, Z,) and its corresponding prices P;» q, regardless of 
whether that path passes the test for optimality. Propositions 
(H), (I) together express that test. 
(F) If (x, 2,) is an eligible and attainable patn, and 1] 
(X,, z,) is any bath, feasible or not, then 
(24) 
Ug. 
l 
A N 
py - Xs) at 
for all finite T. and for 
T 
x whenever the integral con», 
Since both members vanish if x.=4, for all sz, this means 
that both 
(1) the utility Ur(p) of the path (x, z,) 1s maximized, subject 
onlv to the « budget constraint » 
if (a, 
. 
A 
+2 
1 
<,,, and 
- xy Ue 
— 
+1 Koopmans - pag. 2.